The existence of a hypothetical parallel mirror world, with microphysics identical to that of the observable world and connected to the latter only gravitationally, has a non-trivial impact on cosmology (for a review, see e.g. [1]}).
In particular, extra relativistic species in the early Universe (mirror photons, electrons and neutrinos) affect the number of effective neutrino species
\(N_{\rm eff}\)  that can be measured through: i) the abundances of light elements –in particular Helium– at
the time of Big Bang nucleosynthesis (BBN), i.e. at \(T\sim 1\, {\rm MeV}\) ; ii) through the damping tail in the cosmic microwave background (CMB) power spectrum at recombination, \(T\sim 0.26\, {\rm eV}\) .
Present data yield [2]}, [3]}, [4]}For BBN, the constraint corresponds to \(BBN+Yp+D\)  in Table V of Ref. [2]}, translating their 68.3% CL result for the number of neutrino species \(N_\nu \) 
to a 95% CL value \(N_{\nu }=2.85 \pm 0.28\) , and \(N_{\mathrm {eff}}=1.015 \, N_{\nu }\) .
For CMB, the
combination of TT+TE+EE+lowE+lensing+BAO data by Planck 2018 [3]}, [4]} is considered here; note that due to the disagreement between local and CMB measurements of the Hubble expansion parameter \(H_0\) , the constraint on \(\Delta N_{\rm eff}\)  would weaken to \(N_{\mathrm {eff}}=3.27 \pm 0.30\)  if \(H_0\)  measurements were included.
\(\text{BBN: } N_{\text{eff}}=2.89 \pm 0.57 \,\,,\qquad \text{CMB: } N_{\text{eff }}=2.99_{-0.33}^{+0.34}\,.\) 
